Pokémon Ruby Version, Sapphire Version, and Emerald Version - Mirage Island
Once each day, the game generates a random integer from 0 through 65535. Mirage Island will be "visible" (on Route 130, east of Pacifidlog Town) if that random number matches the lower 16 bits of the personality ID of at least one Pokémon in the player’s party when Route 130 is entered.